The Seifert-van Kampen Theorem

Table of Contents

We are now ready to state a very important theorem for determining the fundamental groups of certain spaces. This theorem is called the Seifert-van Kampen theorem. We first state the simplest version of the theorem and then state a more general version.

Theorem 1 (The Seifert-van Kampen Theorem): Let

$X$

be a topological space and let

\{ U_1, U_2 \}

be an open cover of

$X$

. Let

\pi_1(U_1, x) = \langle A_1: \mathcal R_1 \rangle

and

\pi_1(U_2, x) = \langle A_2 : \mathcal R_2 \rangle

. If:
1)

$U_1$

$U_2$

, and

U_1 \cap U_2

are path connected.
2)

x \in U_1 \cap U_2

.
Then

\pi_1(X, x)

is the free product of

\pi_1(U_1, x)

and

\pi_1(U_2, x)

with amalgamation by

\pi_1(U_1 \cap U_2, x)

i_{1*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_1, x)

and

i_{2*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_2, x)

, that is,

\pi_1(X, x) = \langle A_1 \cup A_2 : \mathcal R_1 \cup \mathcal R_2 \cup \{ i_{1*}(h) = i_{2^*}(h) : h \in \pi_1(U_1 \cap U_2, x) \} \rangle

We denote the inclusion map of $U_1 \cap U_2$ in $$U_1$$ by $i_1 : U_1 \cap U_2 \to U_1$ , and we denote the inclusion map of $U_1 \cap U_2$ in $$U_2$$ by $i_2 : U_1 \cap U_2 \to U_2$ . Hence, $i_{1*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_1, x)$ and $i_{2*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_2, x)$ are the induced mapping as seen on The Induced Mapping from the Fundamental Groups of Two Topological Spaces page, and they are also group homomorphisms.

Theorem 2 (The General Seifert-van Kampen Theorem): Let

$X$

be a topological space and let

\{ U_j : j \in J \}

be an open cover of

$X$

. Let

\pi_1(U_j, x) = \langle A_j, \mathcal R_j \rangle

for each

j \in J

. If:
1) Each intersection of four or less

$U_j$

s is path connected.
2)

\displaystyle{x \in \bigcap_{j \in J} U_j}

.
Then

\pi_1(X, x)

is the free product of the

\pi_1(U_j, x)

s with amalgamation by the maps

\phi_{ij*}^{i} : \pi_1(U_i \cap U_j, x) \to \pi_1(U_i, x)

and

\phi_{ij*}^{j} : \pi_1(U_i \cap U_j, x) \to \pi_1(U_j, x)

, that is,

\pi_1(X, x) = \biggr \langle \bigcup_{j \in J} A_j : \bigcup_{j \in J} \mathcal R_j \cup \bigcup_{i, j \in J} \{ \phi_{ij*}^{i}(h) = \phi_{ij*}^{j}(h) : h \in \pi_1(U_i \cap U_j, x) \} \biggr \rangle

Here, $\phi_{ij}^{i} : U_i \cap U_j \to U_i$ and $\phi_{ij}^{j} : U_i \cap U_j \to U_j$ are the usual inclusion maps.

We look at some examples of applying the Seifert-van Kampen theorem on the following pages:

The Seifert-van Kampen Theorem Example 1

The Seifert-van Kampen Theorem Example 2

The Seifert-van Kampen Theorem Example 3

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The Seifert-van Kampen Theorem